The small perturbation Seiberg–Witten conjecture for property $(\ast)$

Let $M$ be a smooth 4-manifold. Assume that $b_2^+(M)=1$, $b_2^-(M)\leq 9$, $H_1(M;\mathbb{Z})=0$, and that $M$ has non-trivial small perturbation Seiberg–Witten invariants. Property $(\ast)$ is the property defined in the surrounding paper.

Small perturbation Seiberg–Witten conjecture. Under these assumptions, $M$ satisfies property $(\ast)$.

The conjecture concerns restrictions on smooth 4-manifolds with $b_2^+=1$ and controlled negative intersection form in the presence of non-trivial Seiberg–Witten invariants. The supplied text does not define property $(\ast)$ or provide evidence resolving the conjecture.